Contents

Idea

A variant of the idea of generalized complex geometry given by passing from generalization of complex geometry to generalization of exceptional geometry. Instead of by reduction of structure groups along inclusions like $O(d)\times O(d)\to O(d,d)$ it is controled by incusions of into split real forms of exceptional Lie groups.

This serves to neatly encode U-duality groups in supergravity as well as higher supersymmetry of supergravity compactifications.

Examples

Higher supersymmetry

Compactification of 11-dimensional supergravity on a manifold of dimension 7 preserves $N=1$ supersymmetry precisely if its generalized tangent bundle has G-structure for the inclusion

of the special unitary group in dimension 7 into the split real form of E7. This is shown in (Pacheco-Waldram).

One dimension down, compactification of 10-dimensional type II supergravity on a 6-manifold $X$ preserves $N=2$ supersymmetry precisely if the generalized tangent bundle $TX\otimes T^{*}X$ in the NS-NS sector admits G-structure for the inclusion

This is reviewed in (GLSW, section 2).

Related concepts

• generalized complex geometry, exceptional geometry

• generalized tangent bundle, exceptional tangent bundle

• generalized Calabi-Yau manifold, generalized G2-manifold

• Bn-geometry, differential T-duality

References

General

• David Baraglia, Leibniz algebroids, twistings and exceptional generalized geometry (arXiv:1101.0856)

In supergravity

Survey slides include

• David Baraglia, Exceptional generalized geometry and $N=2$ backgrounds (pdf)

Reviewes include

• Daniel Persson, Arithmetic and Hyperbolic Structures in String Theory (arXiv:1001.3154)

• Nassiba Tabti, Kac-Moody algebraic structures in supergravity theories (arXiv:0910.1444)

Original articles include

• Chris Hull, Generalised Geometry for M-Theory, JHEP 0707:079 (2007) (arXiv:hep-th/0701203)

• Paulo Pires Pacheco, Daniel Waldram, M-theory, exceptional generalised geometry and superpotentials (arXiv:0804.1362)

• Mariana Graña, Jan Louis, Aaron Sim, Daniel Waldram, $E_{7(7)}$ formulation of $N=2$ backgrounds (arXiv:0904.2333)

• G. Aldazabala,, E. Andrésb, P. Cámarac, Mariana Graña, U-dual fluxes and generalized geometry, JHEP 1011:083,2010 (arXiv:1007.5509)

• Mariana Graña, Francesco Orsi, $N=1$ vacua in Exceptional Generalized Geometry (arXiv:1105.4855)

• Mariana Graña, Francesco Orsi, N=2 vacua in Generalized Geometry, (arXiv:1207.3004)

• André Coimbra, Charles Strickland-Constable, Daniel Waldram, $E_{d(d)}\times {ℝ}^{+}$ Generalised Geometry, Connections and M theory (arXiv:1112.3989)

The E10-geometry of 11-dimensional supergravity compactified to the line is discussed in

• Thibault Damour, Hermann Nicolai, Higher order M theory corrections and the Kac-Moody algebra E10 (arXiv:hep-th/0504153)

The E11-geometry of 11-dimensional supergravity compactified to the point is discussed in

• Peter West, Generalised geometry, eleven dimensions and E11 (arXiv:1111.1642)

The generalized-U-duality+diffeomorphsim invariance in 11d is discussed in

• David S. Berman, Martin Cederwall, Axel Kleinschmidt, Daniel C. Thompson, The gauge structure of generalised diffeomorphisms (arXiv:1208.5884)

For the worldvolume theory of the M5-brane this is discussed in

• Machiko Hatsuda, Kiyoshi Kamimura, M5 algebra and $\mathrm{SO}(5,5)$ duality (arXiv:1305.2258)